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4112 bytes, as of the version from 2026-11-10 22:01 (current). Recorded here, not run.
"""Untapered Möbius approximation errors -- numberdb.org/T706.
Run with SageMath or compatible passagemath and numberdb:
sage -pip install numberdb
sage -python generate.py # verify every stored entry; no write
sage -python generate.py --publish # fill the authorized draft, no offering
For a private draft configure NUMBERDB_API_KEY. No credentials belong here.
The campaign wrapper may provide the key on stdin with NUMBERDB_KEY_FROM_STDIN=1.
U_N = ||chi + sum_(n=1)^N mu(n) rho_n||^2, in L^2((0,infinity),dx),
rho_n(x)={1/(nx)}, chi=1_(0,1). N is a positive integer; default N=1,...,30.
Finite cotangent sums (BBLS, math/0306251v1, Proposition 89) evaluate the
Gram kernel at exact rational ratios. All transcendental operations and norm
arithmetic use Arb balls, at numberdb.bits(digits, losing=128) working bits.
No companion-table decimals are input. See derivation.md for the formulas,
checks.py for the separate integral route, and numerical-evidence.json for
measured enclosure accuracy. The gram_system implementation is reused from
T641's attached generate.py, SHA-256 7e155af1d4a4963f874e0c536051c1cf0b7afd01e21b93d64a674c7fab8592c8,
after inspecting its formula and checking every required kernel independently.
"""
import os
import sys
from functools import lru_cache
from math import gcd
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from sage.arith.misc import moebius
MAX_N = 30
WORKING_GUARD = 128
def gram_system(limit, precision):
"""Fresh finite cotangent sums, Gram entries, and target inner products."""
R = RealBallField(precision)
C = (2*R.pi()).log() - R.euler_constant()
@lru_cache(None)
def vasyunin(p, q):
return sum((R(QQ((m*p) % q)/QQ(q)) *
(R.pi()*R(QQ(m)/QQ(q))).cot()
for m in range(1, q)), R(0))
@lru_cache(None)
def autocorrelation(p, q):
x = R(QQ(p)/QQ(q))
return (((1-x)*x.log() + (1+x)*C)/2
- R.pi()*(vasyunin(p,q)+vasyunin(q,p))/(2*q))
G = [[R(0) for k in range(limit)] for j in range(limit)]
for j in range(1, limit+1):
for k in range(1, j+1):
g = gcd(j,k)
G[j-1][k-1] = G[k-1][j-1] = autocorrelation(j//g,k//g)/j
b = [(R(n).log()+1-R.euler_constant())/n for n in range(1,limit+1)]
return R, G, b
@lru_cache(None)
def computation(limit, digits):
R, G, b = gram_system(limit, numberdb.bits(digits, losing=WORKING_GUARD))
weights = [moebius(n) for n in range(1, limit+1)]
total = R(1)
values = []
for i, w in enumerate(weights):
total += 2*w*b[i] + w*w*G[i][i]
total += 2*w*sum((weights[j]*G[i][j] for j in range(i)), R(0))
assert total.is_finite() and total > 0
assert total.accuracy() >= numberdb.bits(digits)+16
values.append(total)
return R, G, b, weights, values
class UntaperedErrors(numberdb.Generator):
table = 'T706'
parameters = ('N',)
type = 'R'
digits = 100
rigour = 'proven'
files = ('generate.py', 'checks.py', 'integral_kernel.py',
'argument-selection.md', 'derivation.md', 'numerical-evidence.json',
'skill-manifest.json', 'research.md')
def enumerate(self, limit=MAX_N):
for N in range(1, limit+1):
yield {'N': str(N)}
def value(self, params, digits):
N = int(params['N'])
if N < 1:
raise ValueError('N must be a positive integer')
return computation(max(MAX_N, N), digits)[4][N-1]
if __name__ == '__main__':
if os.environ.get('NUMBERDB_KEY_FROM_STDIN') == '1':
numberdb.configure(api_key=sys.stdin.read().strip())
generator = UntaperedErrors()
if '--publish' in sys.argv or os.environ.get('NUMBERDB_PUBLISH') == '1':
print(generator.publish(message=os.environ.get('NUMBERDB_EDIT_MESSAGE',
'Recompute untapered Mobius errors for N=1,...,30')))
else:
report = generator.verify(sample=None)
print(report)
sys.exit(0 if report.ok else 1)